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K-Theory and Characteristic Classes in Topology and Complex Geometry (A Tribute to Atiyah and Hirzebruch)

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K-Theory and Characteristic Classes in Topology and Complex Geometry (A Tribute to Atiyah and Hirzebruch) K-theory and characteristic classes in topology and complex geometry (a tribute to Atiyah and Hirzebruch) Claire Voisin CNRS, Institut de math´ematiquesde Jussieu CMSA Harvard, May 25th, 2021 Plan of the talk I. The early days of Riemann-Roch • Characteristic classes of complex vector bundles • Hirzebruch-Riemann-Roch. Ref. F. Hirzebruch. Topological Methods in Algebraic Geometry (German, 1956, English 1966) II. K-theory and cycle class • The Atiyah-Hirzebruch spectral sequence and cycle class with integral coefficients. • Resolutions and Chern classes of coherent sheaves Ref. M. Atiyah, F. Hirzebruch. Analytic cycles on complex manifolds (1962) III. Later developments on the cycle class • Complex cobordism ring. Kernel and cokernel of the cycle class map. • Algebraic K-theory and the Bloch-Ogus spectral sequence The Riemann-Roch formula for curves • X= compact Riemann surface (= smooth projective complex curve). E ! X a holomorphic vector bundle on X. •E the sheaf of holomorphic sections of E. Sheaf cohomology H0(X; E)= global sections, H1(X; E) (eg. computed as Cˇech cohomology). Def. (holomorphic Euler-Poincar´echaracteristic) χ(X; E) := h0(X; E) − h1(X; E). • E has a rank r and a degree deg E = deg (det E) := e(det E). • X has a genus related to the topological Euler-Poincar´echaracteristic: 2 − 2g = χtop(X). • Hopf formula: 2g − 2 = deg KX , where KX is the canonical bundle (dual of the tangent bundle). Thm. (Riemann-Roch formula) χ(X; E) = deg E + r(1 − g) Sketch of proof Sketch of proof. (a) Reduction to line bundles: any E has a filtration by subbundles Ei such that Ei=Ei+1 is a line bundle. The 3 quantities r, χ and deg are additive under short exact sequences. (b) Reduction to OX : L= holomorphic line bundle on X, x 2 X. Line bundle L(−x) whose sheaf of sections is L ⊗ Ix, with short exact sequence 0 ! L ⊗ Ix !L! Cx ! 0. One has deg L(−x) = deg (L) − 1; χ(X; L(−x)) = χ(X; L) − 1. ) (*) χ(X; L) = χ(X; OX ) + deg L. (c) Serre duality ) χ(X; KX ) = −χ(X; OX ). Formula (*) for KX then gives 2χ(X; OX ) = −deg KX hence χ(X; OX ) = 1 − g. qed • Surfaces. For a holomorphic line bundle L on a projective surface X, one \easily" gets using the Riemann-Roch formula on curves, 2 L −KX ·L (∗∗) χ(X; L) = χ(X; OX ) + 2 : • Serre duality gives χ(X; OX ) = χ(X; KX ); already contained in (**). • Hirzebruch uses the Hodge index theorem + topological formulae for the 2 c1(X) +c2(X) signature ) Noether formula χ(X; OX ) = 12 . Sketch of proof Sketch of proof. (a) Reduction to line bundles: any E has a filtration by subbundles Ei such that Ei=Ei+1 is a line bundle. The 3 quantities r, χ and deg are additive under short exact sequences. (b) Reduction to OX : L= holomorphic line bundle on X, x 2 X. Line bundle L(−x) whose sheaf of sections is L ⊗ Ix, with short exact sequence 0 ! L ⊗ Ix !L! Cx ! 0. One has deg L(−x) = deg (L) − 1; χ(X; L(−x)) = χ(X; L) − 1. ) (*) χ(X; L) = χ(X; OX ) + deg L. (c) Serre duality ) χ(X; KX ) = −χ(X; OX ). Formula (*) for KX then gives 2χ(X; OX ) = −deg KX hence χ(X; OX ) = 1 − g. qed • Surfaces. For a holomorphic line bundle L on a projective surface X, one \easily" gets using the Riemann-Roch formula on curves, 2 L −KX ·L (∗∗) χ(X; L) = χ(X; OX ) + 2 : • Serre duality gives χ(X; OX ) = χ(X; KX ); already contained in (**). • Hirzebruch uses the Hodge index theorem + topological formulae for the 2 c1(X) +c2(X) signature ) Noether formula χ(X; OX ) = 12 . Chern classes of complex vector bundles (Chern, Borel, Borel-Hirzebruch) • Chern. E= complex differentiable vector bundle on a manifold X. 1 C-linear Hermitian connection r on E curvature Rr = 2iπ r ◦ r and k real closed forms Tr Rr of degree 2k real cohomology classes. • Chern classes ck(E) :=\k-th symmetric functions of the eigenvalues of Rr". Related to the classes above by the Newton formulas. 2 • L=complex line bundle on X first Chern class c1(L) 2 H (X; Z). 1 0 ∗ 2 Defined using the map H (X; (C ) ) ! H (X; Z) induced by the 0 0 ∗ exponential exact sequence 0 ! Z !C ! (C ) ! 1. Thm. (Axiomatic construction/characterization of Chern classes) There 2i exist unique Chern classes ci(E) 2 H (X; Z) for any E; X, with total P Chern class c(E) = i ci(E) satisfying the following axioms. (i) Contravariant functoriality. (ii) (Whitney formula) c(E ⊕ F ) = c(E) · c(F ). (iii) c(L) = 1 + c1(L), where c1(L) is as defined above. The proof uses the splitting principle : Given E ! X, there exists a ∗ ∗ ∗ ∗ f : Y ! X such that f : H (X; Z) ! H (Y; Z) is injective and f E is a direct sum of line bundles. Chern classes of complex vector bundles (Chern, Borel, Borel-Hirzebruch) • Chern. E= complex differentiable vector bundle on a manifold X. 1 C-linear Hermitian connection r on E curvature Rr = 2iπ r ◦ r and k real closed forms Tr Rr of degree 2k real cohomology classes. • Chern classes ck(E) :=\k-th symmetric functions of the eigenvalues of Rr". Related to the classes above by the Newton formulas. 2 • L=complex line bundle on X first Chern class c1(L) 2 H (X; Z). 1 0 ∗ 2 Defined using the map H (X; (C ) ) ! H (X; Z) induced by the 0 0 ∗ exponential exact sequence 0 ! Z !C ! (C ) ! 1. Thm. (Axiomatic construction/characterization of Chern classes) There 2i exist unique Chern classes ci(E) 2 H (X; Z) for any E; X, with total P Chern class c(E) = i ci(E) satisfying the following axioms. (i) Contravariant functoriality. (ii) (Whitney formula) c(E ⊕ F ) = c(E) · c(F ). (iii) c(L) = 1 + c1(L), where c1(L) is as defined above. The proof uses the splitting principle : Given E ! X, there exists a ∗ ∗ ∗ ∗ f : Y ! X such that f : H (X; Z) ! H (Y; Z) is injective and f E is a direct sum of line bundles. Chern classes of complex vector bundles (Chern, Borel, Borel-Hirzebruch) • Chern. E= complex differentiable vector bundle on a manifold X. 1 C-linear Hermitian connection r on E curvature Rr = 2iπ r ◦ r and k real closed forms Tr Rr of degree 2k real cohomology classes. • Chern classes ck(E) :=\k-th symmetric functions of the eigenvalues of Rr". Related to the classes above by the Newton formulas. 2 • L=complex line bundle on X first Chern class c1(L) 2 H (X; Z). 1 0 ∗ 2 Defined using the map H (X; (C ) ) ! H (X; Z) induced by the 0 0 ∗ exponential exact sequence 0 ! Z !C ! (C ) ! 1. Thm. (Axiomatic construction/characterization of Chern classes) There 2i exist unique Chern classes ci(E) 2 H (X; Z) for any E; X, with total P Chern class c(E) = i ci(E) satisfying the following axioms. (i) Contravariant functoriality. (ii) (Whitney formula) c(E ⊕ F ) = c(E) · c(F ). (iii) c(L) = 1 + c1(L), where c1(L) is as defined above. The proof uses the splitting principle : Given E ! X, there exists a ∗ ∗ ∗ ∗ f : Y ! X such that f : H (X; Z) ! H (Y; Z) is injective and f E is a direct sum of line bundles. Chern character and Todd genus; formalism of virtual roots • Virtual roots and symmetric functions. For any symmetric polynomial f in k variables λ1; : : : ; λk, one has a polynomial Pf in the symmetric functions σi of λ1; : : : ; λk, such that Pf (σ·) = f(λ·). • Works as well with formal series. If f has coefficients in A, so does Pf . ∗ • E a vector bundle of rank k on X with Chern classes ci(E) 2 H (X; Q). ∗ For any f as above Pf (c·(E)) 2 H (X; Q). The λi implicitly used in the function f are called the virtual roots of the Chern polynomial. When the vector bundle is a direct sum of line bundles, one can take λi = c1(Li). • In general, the λi can be realized as cohomology classes only on a splitting manifold Y ! X for E. P • Chern character: ch E = i exp λi. Obviously ch (E ⊕ F ) = ch E + ch F , ch (E ⊗ F ) = ch E · ch F . • Todd genus. td E = Q λi . Obviously i 1−exp(−λi) td (E ⊕ F ) = td E · td F . Hirzebruch-Riemann-Roch formula • E=complex vector bundle on X= complex manifold. TX has a complex structure Chern classes ci(E); cj(TX ). • Holomorphic structure on E sheaf E of holomorphic sections, cohomology groups Hi(X; E) and holomorphic Euler-Poincar´e P i i characteristic χ(X; E) := χ(X; E) = i(−1) h (X; E) (X compact). Thm. (Hirzebruch-Riemann-Roch formula) One has R χ(X; E) = X ch E · td X =: T0(X; E). • The χy-genus. TX is a holomorphic vector bundle, hence also ∗ P p p ΩX = TX . Define χy(E) := p y χ(X; E ⊗ ΩX ). • Obvious. χ(X; E) = χ0(E). • Less obvious, due to Serre. For the trivial bundle OX , one has χ−1(X; OX ) = χtop(X). n Proof. Holomorphic de Rham complex 0 !OX ! ΩX ! ::: ! ΩX ! 0. This is a resolution of the constant sheaf C. qed • Ty-genus Ty(X; E) : plug-in y in the formal expression for ch E · td X, P eg chy(E) = i exp(1 + y)λi.
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